Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szeg\"o equation

TL;DR

This paper investigates how randomization affects nonlinear smoothing in dispersive PDEs, demonstrating smoothing for the periodic KdV equation but not for the cubic Szeg"o equation, with implications for understanding solution regularity.

Contribution

It constructs local solutions for the periodic KdV with rough initial data using nonlinear smoothing techniques and shows the absence of such smoothing for the cubic Szeg"o equation under randomization.

Findings

Nonlinear smoothing occurs for the periodic KdV under random initial data.

No nonlinear smoothing is observed for the cubic Szeg"o equation with randomized initial data.

Solutions exist locally in time for KdV with initial data in certain rough Sobolev spaces.

Abstract

We consider Cauchy problems of some dispersive PDEs with random initial data. In particular, we construct local-in-time solutions to the mean-zero periodic KdV almost surely for the initial data in the support of the mean-zero Gaussian measures on H^s(T), s > s_0 where s_0 = -11/6 + \sqrt{61}/6 \thickapprox -0.5316 < -1/2, by exhibiting nonlinear smoothing under randomization on the second iteration of the integration formulation. We also show that there is no nonlinear smoothing for the dispersionless cubic Szeg\"o equation under randomization of initial data.